Growth rate for beta-expansions

Let $\beta>1$ and let m>\be be an integer. Each x\in I_\be:=[0,\frac{m-1}{\beta-1}] can be represented in the form $$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$ where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\beta$. When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.Comment: 21 pages, 2 figure